 Pick up of the last lecture. So we consider it a model of tensor category. If we construct it, so given a model of tensor category, construct it this three-dimensional tq of t. And so in particular, what we did is, so we defined what is the value of this functor of this functor on some geno-g Riemann surface, some explicit expression in terms of homes between objects, between some combination of simple objects in this model tensor category. And then we also, by explicit formula, we defined the value on some cobaltism between two Riemann surfaces. And the idea of this was actually tip two, so since it's a functor between some spaces, it's a map between some vector spaces, we just pair it between the corresponding elements, and this pairing corresponds to closure, to certain closure, to certain canonical closure of this manifold. So we obtain a certain close three-dimensional manifold but with certain coupons embedded, and the coupons are labelled by the elements of the vector spaces associated with the boundary. And this is closed, any closed three-manifold with these coupons embedded can be understood as a dense surgery on a certain link in a three together with those coupons. So there's the coupons and the link and interlink with each other. And the formula was essentially as a generalization of the formula for WRT invariant. So in particular, let me make the following remarks. So the remark number one is that, so if I take, so if I want to recover my WRT invariant on a closed three-manifold, so this is given by this general construction where C, so there are two ways to kind of understand the C, at least is, so let me write an inverse as a category of finite dimensional representations. So there is some fine print to it, some extra subtle conditions, finite dimensional representations of quantum SL2. So this is at Q equals two pi I divided by K. So this is some whole algebra. And so this model, the representation theory of this whole algebra is very different for generic Q and Q square to F. And so in particular, but there is a simple way to understand simple, the objects, the simple objects in this category. So as I said, this can be identified with just representations of SL2 of dimensions one, two, two, K minus one. But of course, there is some non-trivial tensor product. Now the tensor product will be modified and we have to put all other some non-trivial, some non-trivial, this MTC structure on those objects. So in particular, so how does the generic MTC data is expressed in this case? So in particular, this quantum dimensions, there will be just the quantum numbers. And so here, again, I label my simple objects by just numbers from one to K minus one, which correspond to the dimensions. And for example, S matrix, these unnormalized S matrix is just, it's a value of, it's a color range of my hope fling. What is this? Or maybe, and so on. Of course, there are also some air matrix and so on, which one can work. So but another way to understand this category is this category, can be ensued as a category of what is called integrable representations of fine, the algebra, SU2 at level K. I would say this, because this is kind of, there's a more general understanding, you can understand this C as a category of modules of some vertex operator algebra. And the vertex operator algebra is what describes you as chiral conformal fields here, which lives on the boundary of this three-dimensional 2KFT. And it's kind of this, in terms of, kind of this rational conformal field theory, this MTC structure was actually constructed by, kind of described by, in the paper by Muran Zaidak. Yes, thank you. So another remark, so here, by this formula, we define the value on this bordism, but we haven't checked actually if the gluing of bordism corresponds to composition of maps. And if one does this, one actually comes to some sort of difficulty. So the gluing is functorial only up to a face. The face, which is, it's in general has some power of a certain ratio where this ratio, these numbers, in terms of MTC data are defined as false. And so this face, this kind of the extra face, so which means when we compose, when we can see the composition of bordisms, we get a corresponding composition of linear maps, but up to an overall face. But, and this is a kind of, the extra face is correspond to what is often called as framing anomaly. And so this can be cured by considering instead this ZC as a functor from the category of not the ordinary bordism, but the category of framed bordisms. And so which means for each three manifold, we need to specify framing. And here by framing is a choice of trivialization of the direct sum of two copies of tangent bundle. But of course in certain cases when this ratio is one, then it's, then defines a pure TQFT, as the ordinance really TQFT. So for, if you can see the close three manifolds, then there's actually some canonical choice of framing, and that's why actually one can define the WRT invariant is just invariant of three manifolds, not frame manifold. But once we want to start gluing those guys, we need to keep track of framing. But the dependence of framing is very simple, with just overall face. Yes, yes, so for all of them we need to do this. Capital T, oh well this is, I defined this in the previous lecture, so if I can see the following part of the diagram, where I label this strand by simple object I, this is the same as the just a strand without this loop multiplied by TI, okay? And so the final remark which I want to make is that, so this construction, well let me write it this way. So an MTCC actually defines what is called three to one extended TQFT, which can be understood as a factor between the following two categories. So this is, so the right hand side is the two category where the objects are linear categories. And so the, and so the one, the one morphisms are functors between them and well the two morphisms are natural transformations. And here so the objects are one manifolds, which is just a bunch of circles. And so the one morphisms between the objects are kind of two dimensional broadisms. So there's a kind of Riemann surfaces with boundaries and then the two morphisms are abord, abordisms between these two manifolds with boundaries, which can be understood as manifolds with corners. And so in particular, so this, as this two category, the function between two categories, so the value on the empty one manifold is just a easy category of vector spaces and the value on the S one is the category C itself. So now I want to change the topic a little bit. Are there any questions so far? Yeah, this is more like, those guys are more like in one to one correspondence between MTCC. Yes, well I guess this is for some standard linear framing, yeah I'm not sure if I'm ready to say what is the, what is the dependence on framing. I think, oh it's a zero. I guess, yes but, yeah I'm not, I don't know, but I think yes, but I don't know what this is. I don't remember what is this, question. I mean roughly speaking is closely related to this framing anomaly, okay. So we started, so at some point we mentioned that there is a, there is a following in a pass integral realization of the double-rate environment, which is written like this. And so the integral over all connections on, so SU2 connections on M3 modular gauge transformations. And so what, so this is ill-defined but can we learn something from this? Can we learn something precise from this? And so in particular can we, so this has a very particular, so the integral has a very particular dependence on K and one can wonder if you can learn something about what is the dependence, in principle this environment is defined for integer K but either some nice analytic properties of this integral with respect to K. And so we know for example that if you have some just ordinary integral of this form, so the asymptotic expansion, so the behavior, so asymptotics at K goes to infinity is determined by a behavior around critical points of F. So can we make, can we use this kind of finite dimensional intuition to make some conjecture about the double-rate environment? And indeed this is the case. So it's usually referred as asymptotic expansion conjecture. So this can be attributed to Witten and there are also many other people who participate in this material. The momentary mathematical approach was kind of formulated by Anderson. So the conjecturer says that, well I think he mentions this already in this John's corner field here in John's polynomial. But I think more, like more mathematically precise was formulated by Anderson. So the statement that this has, so asymptotics of WRT environment has the following form. So before, okay, sorry, before formulating this, well okay, let me write it first and then I explain the notations. So there'll be some over alpha grants over a certain set. So this will be a finite set. And so explain what it is. So this is a model is, as the set of connected components of the model space of SU2-flat connections on M3. So let me explain the various ingredients here. So first of all is, so the critical points of this transimons functional flat connections. So one can check, it's easy to check that the the equation which tells us the duration of transimons functional with respect to connection one form is equivalent to the condition that the curvature, the curvature is zero. And so the set of critical points model gauchan formations is, so by definition it's a set of connections, A is flatness condition divided by gauchan formations and this can be, so this is even though we can see the some critical points in some infinite dimensional space, this set and it's a space of solutions, model gauchan formation is a finite dimension and it can be explicitly described as forms as a space of homes from pi one from the fundamental group of M3. To SU2 divided by conjugate action of SU2 and the correspondence is given as follows. So if I have some flat connection A, then I need to construct this home and the home is given by, so if I take some element of pi one represented by some cycle gamma, I map it to the holonomy of my connection A around gamma. And of course, if I want to do a gauchan formation here, the holonomy changes by a conjugate action of the, so here I need to choose a base point and so the action of the gauchan formation at this base point will give me a conjugate action on this holon. So this is by definition is my space M flat SU2 of M3 and it has a finite number of connected components. So this is a finite set. So this is a kind of a weak form of this conjecture. So the conjecture just says that there is the asymptotic expansion of this form where the kind of the sum, so the asymptotic is determined by the sum of terms. Each term has exponential with respect to k behavior determined by the value of the transimmon's functional on any connection from this component, because of course the value of the transimmon's functional on any connection from the same connected component is the same, just follows from this equation. And then there's also some expansion with respect to one over k, k is the level with some beginning power. And then there's also a kind of stronger version of this conjecture that those coefficients can be actually independently defined. So this, so the coefficients a alpha and what is called as a called perturbative invariance of M3. And so they can be independently, so they can be defined be defined under certain assumptions in terms of certain counting of trivalent graphs in M3. So in physics, this trivalent graph corresponds to some sort of Feynman diagrams and by counting means integrating over vertices of those graphs with some particular weights. So this has been kind of on the mathematical level of rigor this was considered by Konsevich. So this is a work of in the 90s. And then also both Katania and Axelrod Singer. No, they defined this count, they didn't show this. They just defined those guys independently and then you can make a stronger conjecture that those are the same guys which they defined. Well, one of the conditions, so one good condition is a trivial flat connection in the rational homology sphere. Another good condition is that the flat connection is what's called acyclic. So the corresponding differential has trival homology. So in particular, yeah, you won't isolate it. But so this, of course, but this count is in practice very hard to implement but there is another kind of approach, more explicit. So in particular those guys for trivial flat connections can be defined, can be calculated or you can define them in a different way. So here we are surgery and this was done by Otsuki, so the corresponding series also known as Otsuki series and also there was a relevant work by Razanski who also continues. And so the way you compute it is actually, so suppose your manifold is given by some dense surgery, say for simplicity on a note. And the statement that if you actually know first N-calorie-johns polynomials, you can actually, this allows you to produce first N guys here. And another nice case where you can perform kind of explicit kind of combinatorial calculation is that the case when those guys correspond to irreducible flat connection. So irreducible means the stabilizer with respect to SU2 conjugate action is trivial. So if I take a certain flat connection which is beta is irreducible, so this can be calculated in terms of, of triangulation of M3 and there is those corresponding work is by Dimov Gukov and Zagnyar. Well, so first, I mean you can define those guys and then you can formulate a stronger version of this like conjecture, simple exponential conjecture that those are the same. Yes, yes, yes, you also can show that it's an invariant. It's an invariant. Yeah, this is normal, it's not a suki-series invariant. Yes, I don't know, I think yes. Okay, so, but now, any other questions? Now we actually want to, so it's not, it has been verified in many cases, but it's not has not been proven. But actually we want to, so we wanted to make a stronger conjecture. It's not just, we want, so we want to make something stronger, not just say the relation, not just state some conjecture about the relation of these coefficients to